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Theorem exmidontri2or 7220
Description: Ordinal trichotomy is equivalent to excluded middle. (Contributed by Jim Kingdon, 26-Aug-2024.)
Assertion
Ref Expression
exmidontri2or  |-  (EXMID  <->  A. x  e.  On  A. y  e.  On  ( x  C_  y  \/  y  C_  x ) )
Distinct variable group:    x, y

Proof of Theorem exmidontri2or
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 exmidontriim 7202 . . 3  |-  (EXMID  ->  A. x  e.  On  A. y  e.  On  ( x  e.  y  \/  x  =  y  \/  y  e.  x ) )
2 onelss 4372 . . . . . . . 8  |-  ( y  e.  On  ->  (
x  e.  y  ->  x  C_  y ) )
32adantl 275 . . . . . . 7  |-  ( ( x  e.  On  /\  y  e.  On )  ->  ( x  e.  y  ->  x  C_  y
) )
4 orc 707 . . . . . . 7  |-  ( x 
C_  y  ->  (
x  C_  y  \/  y  C_  x ) )
53, 4syl6 33 . . . . . 6  |-  ( ( x  e.  On  /\  y  e.  On )  ->  ( x  e.  y  ->  ( x  C_  y  \/  y  C_  x ) ) )
6 eqimss 3201 . . . . . . . 8  |-  ( x  =  y  ->  x  C_  y )
76, 4syl 14 . . . . . . 7  |-  ( x  =  y  ->  (
x  C_  y  \/  y  C_  x ) )
87a1i 9 . . . . . 6  |-  ( ( x  e.  On  /\  y  e.  On )  ->  ( x  =  y  ->  ( x  C_  y  \/  y  C_  x ) ) )
9 onelss 4372 . . . . . . . 8  |-  ( x  e.  On  ->  (
y  e.  x  -> 
y  C_  x )
)
109adantr 274 . . . . . . 7  |-  ( ( x  e.  On  /\  y  e.  On )  ->  ( y  e.  x  ->  y  C_  x )
)
11 olc 706 . . . . . . 7  |-  ( y 
C_  x  ->  (
x  C_  y  \/  y  C_  x ) )
1210, 11syl6 33 . . . . . 6  |-  ( ( x  e.  On  /\  y  e.  On )  ->  ( y  e.  x  ->  ( x  C_  y  \/  y  C_  x ) ) )
135, 8, 123jaod 1299 . . . . 5  |-  ( ( x  e.  On  /\  y  e.  On )  ->  ( ( x  e.  y  \/  x  =  y  \/  y  e.  x )  ->  (
x  C_  y  \/  y  C_  x ) ) )
1413ralimdva 2537 . . . 4  |-  ( x  e.  On  ->  ( A. y  e.  On  ( x  e.  y  \/  x  =  y  \/  y  e.  x
)  ->  A. y  e.  On  ( x  C_  y  \/  y  C_  x ) ) )
1514ralimia 2531 . . 3  |-  ( A. x  e.  On  A. y  e.  On  ( x  e.  y  \/  x  =  y  \/  y  e.  x )  ->  A. x  e.  On  A. y  e.  On  ( x  C_  y  \/  y  C_  x ) )
161, 15syl 14 . 2  |-  (EXMID  ->  A. x  e.  On  A. y  e.  On  ( x  C_  y  \/  y  C_  x ) )
17 ontri2orexmidim 4556 . . . 4  |-  ( A. x  e.  On  A. y  e.  On  ( x  C_  y  \/  y  C_  x )  -> DECID  z  =  { (/)
} )
1817adantr 274 . . 3  |-  ( ( A. x  e.  On  A. y  e.  On  (
x  C_  y  \/  y  C_  x )  /\  z  C_  { (/) } )  -> DECID 
z  =  { (/) } )
1918exmid1dc 4186 . 2  |-  ( A. x  e.  On  A. y  e.  On  ( x  C_  y  \/  y  C_  x )  -> EXMID )
2016, 19impbii 125 1  |-  (EXMID  <->  A. x  e.  On  A. y  e.  On  ( x  C_  y  \/  y  C_  x ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 703  DECID wdc 829    \/ w3o 972    = wceq 1348    e. wcel 2141   A.wral 2448    C_ wss 3121   (/)c0 3414   {csn 3583  EXMIDwem 4180   Oncon0 4348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-uni 3797  df-tr 4088  df-exmid 4181  df-iord 4351  df-on 4353  df-suc 4356
This theorem is referenced by:  onntri52  7221
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